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Quantum Reports

Quantum Work from a Pseudo-Hermitian Hamiltonian

M. Reboiro, Diego Tielas

Published Dec 13, 2022• 2 citations
Publisher

Abstract

In this work, we study the thermodynamics of a hybrid system based on the Da Providencia–Schütte Hamiltonian. The model consists of bosons, i.e., photons in a cavity, interacting with an ensemble of spins through a pseudo-Hermitian Hamiltonian. We compute the exact partition function of the system, and from it, we derive the statistical properties of the system. Finally, we evaluate the work that can be extracted from the system by performing an Otto cycle and discuss the advantages of the proposed pseudo-Hermitian interaction.

DOI
10.3390/quantum4040043

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In this work, we study the thermodynamics of a hybrid system based on the Da Providencia–Schütte Hamiltonian. The model consists of bosons, i.e., photons in a cavity, interacting with an ensemble of spins through a pseudo-Hermitian Hamiltonian. We compute the exact partition function of the system, and from it, we derive the statistical properties of the system. Finally, we evaluate the work that can be extracted from the system by performing an Otto cycle and discuss the advantages of the proposed pseudo-Hermitian interaction.
In study of pseudo(quasi)-hermitian operators, the key role is played by the positive-definite metric operator. It enables physical interpretation of the considered systems. In the article, we study the pseudo-hermitian systems with constant number of particles in equilibrium. We show that the explicit knowledge of the metric operator is not essential for study of thermodynamic properties of the system. We introduce a simple example where the physically relevant quantities are derived without explicit calculation of either metric operator or spectrum of the Hamiltonian.
In the study of pseudo(quasi)-hermitian operators, the key role is played by the positive-definite metric operator. It enables physical interpretation of the considered systems. In this paper, we study the pseudo-hermitian systems with constant number of particles in equilibrium. We show that the explicit knowledge of the metric operator is not essential for the study of thermodynamic properties of the system. We introduce a simple example where the physically relevant quantities are derived without explicit calculation of either metric operator or spectrum of the Hamiltonian.
We use the spin-boson model to describe the dynamics of a two-level atom interacting with Fabry-Pérot cavity modes. We solve the Schrödinger equation for the system-bath model without the Born-Markov approximation to derive the non-Markovian reduced dynamics of the qubit. We further construct an exact Lindblad-type master equation for it. Similar to the quantum Otto cycle, we construct a non- Markovian quasi-cyclic process based on the atom-cavity interactions, which we call the quasi-Otto cycle. For judicious choices of input state and parameters, the quasi-cycle can be more efficient as a quantum engine than the Otto cycle. We also showed that if the quasi-cycle is repeated multiple times, the efficiency of the quasi-Otto engine asymptotically approaches that of the Otto engine.
We study the time evolution of quantum systems with a time-dependent non-Hermitian Hamiltonian exhibiting a SU(1,1) and SU(2) dynamical symmetry. With a time-dependent metric, the pseudo-Hermitian invariant operator is constructed in the same manner as for both the SU(1,1) and SU(2) systems. The exact common solutions of the Schrödinger equations for both the SU(1,1) and SU(2) systems are obtained in terms of eigenstates of the pseudo-Hermitian invariant operator. Finally some interesting physical applications are suggested and discussed.
Attempts to understand the quantum mechanics of non-Hermitian Hamiltonian systems can be traced back to the early days, one example being Heisenberg's endeavour to formulate a consistent model involving an indefinite metric. Over the years non-Hermitian Hamiltonians whose spectra were believed to be real have appeared from time to time in the literature, for instance in the study of strong interactions at high energies via Regge models, in condensed matter physics in the context of the XXZ-spin chain, in interacting boson models in nuclear physics, in integrable quantum field theories as Toda field theories with complex coupling constants, and also very recently in a field theoretical scenario in the quantization procedure of strings on an AdS 5 x S 5 background. Concrete experimental realizations of these types of systems in the form of optical lattices have been proposed in 2007. In the area of mathematical physics similar non-systematic results appeared sporadically over the years. However, intensive and more systematic investigation of these types of non- Hermitian Hamiltonians with real eigenvalue spectra only began about ten years ago, when the surprising discovery was made that a large class of one-particle systems perturbed by a simple non-Hermitian potential term possesses a real energy spectrum. Since then regular international workshops devoted to this theme have taken place. This special issue is centred around the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics held in July 2007 at City University London. All the contributions contain significant new results or alternatively provide a survey of the state of the art of the subject or a critical assessment of the present understanding of the topic and a discussion of open problems. Original contributions from non-participants were also invited. Meanwhile many interesting results have been obtained and consensus has been reached on various central conceptual issues in the growing community of this subject. It is, for instance, well understood that the reality of the spectrum can be attributed either to the unbroken PT-symmetry of the entire system, that is, invariance of the Hamiltonian and the corresponding wavefunctions under a simultaneous parity transformation and time reversal, or more generally to its pseudo-Hermiticity . When the spectrum is real and discrete the Hamiltonian is actually quasi-Hermitian, with a positive-definite metric operator, and can in principle be related by a similarity transformation to an isospectral Hermitian counterpart. For all approaches well-defined procedures have been developed, which allow one to construct metric operators and therefore a consistent description of the underlying quantum mechanical observables. Even though the general principles have been laid out, it remains a challenge in most concrete cases to implement the entire procedure. Solvable models in this sense, some of which may be found in this issue, remain a rare exception. Nonetheless, despite this progress some important questions are still unanswered. For instance, according to the current understanding the non-Hermitian Hamiltonian does not uniquely define the physics of the system since a meaningful metric can no longer be associated with the system in a non-trivial and unambiguous manner. A fully consistent scattering theory has also not yet been formulated. Other issues remain controversial, such as the quantum brachistochrone problem, the problem of forming a mixture between a Hermitian and non-Hermitian system, the new phenomenological possibilities of forming a kind of worm-hole effect, etc. We would like to acknowledge the financial support of the London Mathematical Society, the Institute of Physics, the Doppler Institute in Prague and the School of Engineering and Mathematical Science of City University London. We hope this special issue will be useful to the newcomer as well as to the expert in the subject. Participants of the 6th International Workshop on Pseudo-Hermitian Hamiltonians in Quantum Physics.
The relevance in physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics, and quantum field theory. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of H is obtained using pseudo-bosonic operators.